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J.Aust. Math. Soc. 80 (2006), 419^39

FREE ALGEBRAS IN VARIETIES OF BL-ALGEBRASGENERATED BY A BLn-CHAIN

MANUELA BUSANICHE" and ROBERTO CIGNOLI

(Received 27 November 2003; revised 17 March 2005)

Communicated by B. A. Davey

Abstract

Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by oneBL-chain that is an ordinal sum of a finite MV-chain Ln and a generalized BL-chain B are describedin terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln and freealgebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stonespaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generatedbyLn.

2000 Mathematics subject classification: primary 03G25,03B50, 03B52, 03D35, 03G25, 08B20.Keywords and phrases: BL-algebras, residuated lattices, hoops, Moisil algebras, free algebras, Booleanproducts.

Introduction

Basic Fuzzy Logic (BL for short) was introduced by Hajek (see [19] and the referencesgiven there) to formalize fuzzy logics in which the conjunction is interpreted by acontinuous t-norm on the real segment [0, 1] and the implication by its correspondingadjoint. He also introduced BL-algebras as the algebraic counterpart of these logics.BL-algebras form a variety (or equational class) of residuated lattices [19]. Moreprecisely, they can be characterized as bounded basic hoops [1,7]. Subvarieties ofthe variety of BL-algebras are in correspondence with axiomatic extensions of BL.Important examples of subvarieties of BL-algebras are MV-algebras (that correspondto Lukasiewicz many-valued logics, see [14]), linear Heyting algebras (that correspondto the superintuitionistic logic characterized by the axiom (P =» (2) v (G => P),see [25] for a historical account about this logic), PL-algebras (that correspond to the

© 2006 Australian Mathematical Society 1446-7887/06 $A2.00 + 0.00

419

420 Manuela Busaniche and Roberto Cignoli [2]

logic determined by the t-norm given by the ordinary product on [0, 1], see [15]), andalso Boolean algebras (that correspond to classical logic).

Since the propositions under BL equivalence form a free BL-algebra, descriptions offree algebras in terms of functions give concrete representations of these propositions.Such descriptions are known for some subvarieties of BL-algebras. The best knownexample is the representation of classical propositions by Boolean functions. FreeMV-algebras have been described in terms of continuous piecewise linear functionsby McNaughton [22] (see also [14]). Finitely generated free linear Heyting algebraswere described by Horn [20], and a description of finitely generated free PL-algebraswas given in [15]. Linear Heyting algebras and PL-algebras are examples of varietiesof BL-algebras satisfying the Boolean retraction property. Free algebras in thesevarieties were completely described in [17].

In [10] the first author described the finitely generated free algebras in the varietiesof BL-algebras generated by a single BL-chain which is an ordinal sum of a finiteMV-chain Ln and a generalized BL-chain B. We call these chains BLn-chains. Theaim of this paper is to extend the results of [10] considering the case of infinitely manyfree generators. The results of [10] were heavily based on the fact that the Booleansubalgebras of finitely generated algebras in the varieties generated by BLn-chains arefinite. Therefore the methods of [10] cannot be applied to the general case.

As a preliminary step we characterize the Boolean algebra of idempotent elementsof a free algebra in A4 Vn, the variety of MV-algebras generated by the finite MV-chainLn. It is the free Boolean algebra over a poset which is the cardinal sum of chains oflength n — 1. In the proof of this result a central role is played by the Moisil algebrareducts of algebras in MVn.

Free algebras in varieties of BL-algebras generated by a single BLn-chain Ln l+J Bare then described in terms of weak Boolean products of BL-algebras that are ordinalsums of subalgebras of Ln and free algebras in the variety of basic hoops generatedby B. The Boolean products are taken over the Stone spaces of the Boolean algebrasof idempotent elements of free algebras in MVn. An important intermediate stepis the characterization of the variety of generalized BL-algebras generated by B(Corollary 3.5).

The paper is organized as follows. In the first section we recall, for further reference,some basic notions on BL-algebras and on the varieties MVn. We also recall somefacts about the representation of free algebras in varieties of BL-algebras as weakBoolean products. The only new result is given in Theorem 1.5. In Section 2, aftergiving the necessary background on Moisil algebra reducts of algebras in MVn, wecharacterize the Boolean algebras of idempotent elements of free algebras in MVn.These results are used in Section 3 to give the mentioned description of free algebrasin the varieties of BL-algebras generated by a BLn-chain. Finally in Section 4 we givesome examples and we compare our results with those of [10] and [17].

[3] Free algebras 421

1. Preliminaries

1.1. BL-algebras: basic notions A hoop [7] is an algebra A = (A, *, —>, T) of type(2, 2, 0), such that {A, *, T) is a commutative monoid and for all x, y, z e ^4:

(1) * - + JC = T,(2) x * (x -> y) = y * (y - • * ) ,(3) x -»• (y -> z) = (x * y) ->• z.

A basic /loop [1] or a generalized BL-algebra [18], is a hoop that satisfies the equation

(1.1) ( ( ( * - • > ' ) - • z) * ( ( } ' - • * ) - • z ) ) - > z = T.

It is shown in [1] that generalized BL-algebras can be characterized as algebrasA = (A, A, v, *, -*, T) of type (2 ,2 ,2 ,2 ,0 ) such that

(1) (A, *, T) , is an commutative monoid,(2) L(A) := (A, A, v, T), is a lattice with greatest element T,(3) x — x = T,(4) x -»• (y -> z) = (JC * y) -» z,(5) j[A)i = jt*(j;->y),

(6) U^y)v(y->;t) = T.

A BL-algebra or bounded basic hoop is a bounded generalized BL-algebra, thatis, it is an algebra A = (A, A, v, *, ->•, _L, T) of type (2, 2, 2, 2, 0, 0) such that(A, A, v, *, —•, T) is a generalized BL-algebra, and ± is the lower bound of L(A).In this case, we define the unary operation -> by the equation ->x — x —> ±. TheBL-algebra with only one element, that is, _L = T, is called the trivial BL-algebra.The varieties of BL-algebras and of generalized BL-algebras will be denoted by BCand QBC, respectively.

In every generalized BL-algebra A we denote by < the (partial) order defined on Aby the lattice L(A), that is, for a, b e A, a < b if and only if a — a A b if and only ifb = avb. This order is called the natural order of A. When this natural order is total(that is, for each a,b, e A, a < b or b < a), we say that A is a generalized BL-chain(BL-chain in case A is a BL-algebra). The following theorem makes obvious theimportance of BL-chains and can be easily derived from [19, Lemma 2.3.16].

THEOREM 1.1. Each BL-algebra is a subdirect product of BL-chains.

In every BL-algebra A we define a binary operation x © y = ->(->x * -• v). For eachpositive integer k, the operations xk and k x are inductively defined as follows:

(a) JC' = x and**"1"1 = xk *x,(b) Ix = x and (k + \)x = (kx) © x.


MV-algebras, the algebras of Lukasiewicz infinite-valued logic, form a subvarietyof BC, which is characterized by the equation -•-•jc = x (see [19]). The variety ofMV-algebras is denoted by MV. Totally ordered MV-algebras are called MV-chains.For each BL-algebra A, the set

MV(A) := {x € A :--x = x}

is the universe of a subalgebra MV(A) of A which is an MV-algebra (see [18]).A PL-algebra is a BL-algebra that satisfies the two axioms:

(1) (-—z * ((x * z) -*• (y * z))) -* (x ->• y) = T,(2) x A ->x = _L

PL-algebras correspond to product fuzzy logic, see [15] and [19].It follows from Theorem 1.1 that for each BL-algebra A the lattice L(A) is distrib-

utive. The complemented elements of L(A) form a subalgebra B(A) of A which is aBoolean algebra. Elements of B(A) are called Boolean elements of A.

1.2. Implicative filters

DEFINITION 1.2. An implicative filter of a BL-algebra A is a subset F c. A satisfy-ing the conditions

(1) TeF.(2) If JC € F and x -* y <= F, then y e F.

An implicative filter is called proper provided that F ^ A. If W is a subset of aBL-algebra A, the implicative filter generated by W will be denoted by (W). If U isa filter of the Boolean subalgebra B(A), then the implicative filter (U) is called Stonefilter of A. An implicative filter F of a BL-algebra A is called maximal if and only ifit is proper and no proper implicative filter of A strictly contains F.

Implicative filters characterize congruences in BL-algebras. Indeed, if F is animplicative filter of a BL-algebra A it is well known (see [19, Lemma 2.3.14]), thatthe binary relation =F on A defined by

x =F y if and only if x -*• y e F and y —> x € F

is a congruence of A. Moreover, F = {x e A : x ^F T}. Conversely, if = is acongruence relation on A, then [x € A : x = T} is an implicative filter, and x = y ifand only if x -» y = T and y —> x = T. Therefore, the correspondence F I - » = F isa bijection from the set of implicative filters of A onto the set of congruences of A.

LEMMA 1.3 (see [17]). Let A be a BL-algebra, and let F be a filter o/B(A).Then (=F) = {(a, b)€AxA:a/\c = bA c for some c € F] is a congruencerelation on A that coincides with the congruence relation given by the implicativefilter (F) generated by F.

[5] Free algebras

1.3. MVn-algebras For n > 2, we define:

0 1 2

423

n — I ' M — I n — 1

n- 1

n- 1

The set Ln equipped with the operations x * y = max(0, x + y — 1), x - » y =min(l, 1 — x + y), and with _L = 0 defines a finite MV-algebra, which shall bedenoted by Ln. Clearly fi(Ln) = {0, 1}.

A BL-algebra A is said to be simple provided it is nontrivial and the only properimplicative filter of A is the singleton {T}. In [14], it is proved that Ln is a simpleMV-algebra for each integer n.

We shall denote by M.Vn the subvariety of MV generated by hn. The elementsof MVn are called MVn-algebras. A finite MV-chain Lm belongs to MVn if andonly if An — 1 is a divisor of n — 1. Therefore it is not hard to corroborate that everyMVn-algebra is a subdirect product of a family of algebras (Lm. ,i e I) where mt — 1divides n — 1 for each i e I.

It can be deduced from [14, Corollary 8.2.4 and Theorem 8.5.1] that MVn is theproper subvariety of M. V characterized by the equations

(«„) =x",

and if n > 4, for every integer p = 2, ..., n - 2 that does not divide n - 1

(A,) (pxp-l)n=nxp.

If A is an MVn-algebra, it is not hard to verify that for each x e A\{T},x" = ±and for each y e A \ {J_}, ny = T.

1.4. Ordinal sum and decomposition of BL-chains Let R = (R, *R, -^ R , T) andS = (5, *s, ->s, T) be two hoops such that R D S — {T}. Following [7] we can definethe ordinal sum R W S o f these two hoops as the hoop given by (R U S, *, -» , T)where the operations (*, —•) are defined as follows:

x * y =

y =

x*Ry if x,yeR,

x *s y if x, y e S,

x if x 6 R \ {T} and y e S,

y if y e R \ [T] and x € S.

T if * € fl\{T}, ye S,

x-+Ry \fx,yeR,

x -> s y if *,)> e S,

y if y € R \ [T] and * 6 5.


If R fl 5 ^ {T}, R and S can be replaced by isomorphic copies whose intersectionis {T}, thus their ordinal sum can be defined. When R is a generalized BL-chainand S is a generalized BL-algebra, the hoop resulting from their ordinal sum satisfiesequation (1.1). Thus R l±l S is a generalized BL-algebra. Moreover, if R is a BL-chain,then R l±) S is a BL-algebra, where ± = -LR. If 5 is totally ordered it is obvious thatthe chain R l±l S is subdirectly irreducible if and only if S is subdirectly irreducible.Notice also that for any generalized BL-algebra S, L2 W S is the BL-algebra that arisesfrom adjoining a bottom element to S.

Given a BL-algebra A, we can consider the set D(A) := {x € A : ->x = _L}. It isshown in [18], that D(A) = (D(A), A, v , *,-»-, T) is a generalized BL-algebra.

THEOREM 1.4 (see [10]). For each BL-chain A, we have that A = MV(A) ttlD(A).

THEOREM 1.5. Let Abe a BL-algebra such that MV(A) = L n /or some integer n.Then A = MV(A) W D(A) = Ln l±) D(A).

PROOF. From Theorem 1.1, we can think of each non trivial BL-algebra A as asubdirect product of a family (A,, i e I) of non trivial BL-chains, that is, there existsan embedding e : A —> F]. e / A;, such that 7r,(e(A)) = A, for each i e I, where n,denotes each projection. We shall identify A with e(A). Then each element of A is atuple x and coordinate i is x, 6 At. With this notation we have that for each x € A,7r, (x) = x,. We will prove the following items:

(1) For each i € /, MV(A,) is isomorphic to Ln.

Since for each ( € /, nt is a homomorphism and 7r,(M V(A)) C A,, we have thatn,(MV(A)) c M V(A,). Then 7r,(MV(A)) is a subalgebra of MV(A,). On the otherhand, given i € / , let x, € MV(A,). Then -•-% = x, and there exists an elementx € A such that 7r,(x) = JC,. Taking y = ->-^x e MV(A) we have that 7r,-(y) = x, andxt 6 T T , ( M V ( A ) ) . Hence MV(A,) c 7r,(MV(A)).

In conclusion MV(A,) = 7T , (MV(A) ) = 7r,(Ln) = Ln , because Ln is sim-pie.

(2) Ifx e A, then x € MV(A) U D(A).

Let x e A and let y = n(->x). Ifx, e Ln\ {T}, then ->*,• e Ln\ {J_}. Fromequation (an) we obtain that yt = n(-"X,) = T. On the other hand, if ->*,• = J_,then y, = «(-•*,•) = J_. Now let z = (-•-•x)". Ifx, e Ln\ {T}, then z, = -L, but if->->Xj = T , then zt = T .

Suppose there exists x 6 A such that x £ M V(A) and x £ D(A). It follows fromTheorem 1.4 that for each / 6 / , A, = MV(A,) W D(A,). Then there exist i, j e / ,such that x, € MV(A,) \ {T} = Ln \ {T} andxy e D(A7-) \ {T}.

Let y = n(- 'x). Then _y, = T, y} — _L, and yk € {-L, T} for each k e I \ {i, j}.Now let z = (->->x)". We have that zj — T, z,- = J., and zk e {_L, T} for each


k e I \ {i, j}. It follows that y and z are elements in the chain MV(A) = Ln, whichare not comparable, and this is a contradiction.

(3) lfx e MV(A) \ {T} andy e D(A), then x < y.

The statement is clear if x,•• e M V(A,) \ {T} for every i G / or if yt = T for eachi G / . Otherwise, suppose JC, = T for some i e I. Since x ^ T, there must existj e / such that Xj ^ T. If v, = T for each j e I such that xf = T, then x < y.If not, let z = x A y. Since operations are coordinatewise, Zj € MV(Aj) \ {T} andZt e D(Ai) \ {T}, for some i e I. Hence z £ MV(A) and z ^ D(A), contradictingthe previous item.

(4) lfx e MV(A) \ [T] andy e D(A), then y -> x = xandy * x = x.

Since ->y = _L we have that

y -> x = y ->• -1--X = y ->• ( - * -> 1 ) = --x - ^ (y -> ± )

= --x ->• _L = -•-•x = x,

and

x = y A x = y * ( y — > x) — y *x.

From the previous items it follows that A = MV(A) W D(A) = Ln l±l D(A) . D

1.5. Free algebras in varieties of BL-algebras generated by a BL,,-chain Recallthat an algebra A in a variety JC is said to be free over a set Y if and only if for everyalgebra C in K. and every function / : Y —>• C, / can be uniquely extended to ahomomorphism of A into C. Given a variety K, of algebras, we denote by Free*;(X)the free algebra in /C over X. As mentioned in the introduction, we define a BLn-chainas a BL-chain that is an ordinal sum of the MV-chain Ln and a generalized BL-chain.Once we fixed the generalized BL-chain B, we study the free algebra Freey(X),where V is the variety of BL-algebras generated by the BLn-chain

Tn := Ln W B.

Notice that MV(TB) = Ln and if x £ MV(Tn) \ {T}, then x e D(TH) = B.Recall that a weak Boolean product of a family (Av, y e Y) of algebras over a

Boolean space Y is a subdirect product A of the given family such that the followingconditions hold:

(1) If a, b e A, then [a = b] — {y e Y : ay = by) is open.(2) If a, b e A and Z is a clopen in X, then a\z U£|X\z £ A.

Since the variety Z?£ is congruence distributive, it has the Boolean Factor Congru-ence property. Therefore each nontrivial BL-algebra can be represented as a weakBoolean product of directly indecomposable BL-algebras (see [5] and [23]). The


explicit representation of each BL-algebra as a weak Boolean product of directlyindecomposable algebras is given in [17] by the following lemma.

LEMMA 1.6. Let A be a BL-algebra and let SpB(A) be the Boolean space ofultrafilters of the Boolean algebra B(A). The correspondence

gives an isomorphism of A onto the weak Boolean product of the family

(A/(t/)):C/eSpB(A)

over the Boolean space Sp B(A). This representation is called the Pierce representa-tion. Any other representation of A as a weak Boolean product of a family of directlyindecomposable algebras is equivalent to the Pierce representation.

Therefore, to describe Freey(A') we need to describe B(Freey(X)) and the quo-tients Freev(X)/(U) for each U e SpB(Freev(X)).

In Section 2 we obtain a characterization of the Boolean algebra B(Freey(X)).Once this aim is achieved, we consider the quotients Freev(X)/(f/).

2. B(Freev(A:))

The next two results can be found in [18].

THEOREM 2.1. For each BL-algebra A, B(A) = B(MV(A)).

THEOREM 2.2. For each variety K of BL-algebras and each set X

MV(Free;t(X)) = FneMVnlc(^X).

THEOREM 2.3. V n MV is the variety MVn.

PROOF. Since Ln = MV(Tn) is in V n MV, we have that MVn c V n MV. Onthe other hand, let A be an MV-algebra in V n MV. Suppose A is not in MVn. Thenthere exists an equation e(x\,..., xp) = T that is satisfied by L,, and is not satisfiedby A, that is, there exist elements ait ..., ap in A such that e{ax,..., ap) £ T. Since(->->&i,..., -*->bp) is in (Ln)

p, for each tuple (bu ..., bp) in (Tn)p, the equation

e'{xx,..., xp) = e(->->X\,..., —•—>JCP) = T is satisfied in V. Since A € V n MV,it f o l l o w s t h a t T = e'(au ... ,ap) = e ( - ^ - > a 1 , . . . , -^^ap) — e(au ... ,ap) £ T , a

contradiction. Hence MVn = V n MV. U

From these results we obtain the following theorem.

THEOREM 2.4. B(Freev(X)) = B(FreeMVn(--^X)).


2.1. n-valued Moisil algebras Boolean elements of Freely* (~'~1^0 depend onsome operators that can be denned on each MVn-algebra. Such operators provide eachMVn-algebra with an n-valued Moisil algebra structure, in the sense of the followingdefinition.

DEFINITION 2.5. For each integer n > 2, an n-valued Moisil algebra ([8, 11]) orn-valued Lukasiewicz algebra ([4, 12, 13]) is an algebra

A = (A,A, v,-i,<71",.. . ,or;_,,0, 1)

of type (2, 2, 1 , . . . , 1, 0, 0) such that (A, A, v, 0, 1) is a distributive lattice withunit 1 and zero 0, and ->,&",..., o"_x are unary operators denned on A that satisfythe following conditions:

(1) — x=x,(2) - .(* v y) = -x A ->y,(3) a?(xv y)=cr?xva?y,(4) o?x V -.CT/'JC = 1,

(5) a?ojx = ajx, for i, j = 1, 2 , . . . n - 1,(6) o i-(-x) = -(orB"_,.Jt),(7) ofx V cr,"+1jc = a"+lx, for z = 1, 2, . . . , n - 2,(8) xVaZ_lx=aZ_lx,(9) (* A -io/jc A cr,"+,;y) v j = y, for i = 1, 2 , . . . , n - 2.

Properties and examples of rc-valued Moisil algebras can be found in [4] and [8].The variety of n-valued Moisil algebras will be denoted M.n.

THEOREM 2.6 (see [11]). Let A be in Mn. Then x e B(A) if and only if

Furthermore,

a^_}(x) = min{b e B(A) : x < b] and CT,"(X) = max{a € B(A) : a < x}.

DEFINITION 2.7. For each integer n > 2, a Post algebra of order n is a system

A = (A, A, v , - , CT,\ . . . , < _ , , eu ..., «„_,, 0, 1)

such that (A, A, v, - \ cr" , . . . , CT^,, 0, 1) is an n-valued Moisil algebra and e,,£„_! are constants that satisfy the equations:

(O if i + j <n,


For every n > 2, we can define one-variable terms <J"(X), ..., o£_x(x) in thelanguage (-•, -*, T) such that evaluated on the algebras Ln give

\(n - 1)/ [0 if i+ j < n,

for i = 1, . . . , n - 1 (see [13] or [24]). It is easy to check that

M(LB) = (Ln, A, v, - , af, . . . , a;_,, 0, 1)

is a ^-valued Moisil algebra. Since these algebras are defined by equations and Ln

generates the variety MVn, we have that each A € MVn admits a structure of an«-valued Moisil algebra, denoted by M(A). The chain M(Ln) plays a very importantrole in the structure of n -valued Moisil algebras, since each n -valued Moisil algebrais a subdirect product of subalgebras of M(Ln) (see [4] or [12]). If we add to thestructure M(Ln) the constants e, = i/(n — 1), for i = 1 , . . . , n - 1, then PT(Ln) =(Ln, A, v, --, a", ..., o^_x, eXl ..., en-\, 0, 1) is a Post algebra.

Not every n-valued Moisil algebra has a structure of MVn-algebra (see [21]). Forexample, a subalgebra of M(Ln) may not be a subalgebra of Ln as MVn-algebra. Forinstance, the set

,4 ' 4 ' 4 ' 4

is the universe of a subalgebra of M(L5), but not the universe of a subalgebra of L5.On the other hand, every Post algebra has a structure of MVn-algebra (see [24,Theorem 10]).

The next example will play an important role in what follows.

EXAMPLE 2.8. Let C = (C, A, V, ->, 0, 1) be a Boolean algebra. We define

C 1 " 1 : = [ z = ( Z l , . . . , zn-i) € C"-] :ziSz2<.-.< Zn-i).

For each z = (zi zn-\) e C1"1, we define

0 = (0, . . . . 0),

1 = 0 l),a"(z) = (z,•, z,•,..., zi) for i = 1, n-\.

With A and v defined coordinatewise, C1"1 = (C1"1, A, V, --„, erf,..., an"_,, 0, 1)is an H-valued Moisil algebra (see [8, Chapter 3, Example 1.10]). If we define

f 0 if i < j ,


then C w = (CM, A, v, -.„, a';,..., a ^ . e i , . . . , en_i, 0, 1) is a Post algebra. Con-sequently, C'"1 has a structure of MVn-algebra.

It is easy to see that for each MVn-algebra A, B(A) = B(M(A)).We need to show that the Boolean elements of the MVn-algebra generated by a

set G coincide with the Boolean elements of the n-valued Moisil algebra generatedby the same set. In order to prove this result it is convenient to consider the followingoperators on each n-valued Moisil algebra A. For each i — 0, . . . , n — 1,

where a^(x) = 0 and o^{x) = 1. In M(Ln) we have

j

( n - D

LEMMA 2.9. Let Abe an MVn-algebra, and let G C A. If (G)Mv,, is the subalgebraof A generated by the set G and {G)M,, is the subalgebra o/M(A) generated by G,thenB((G)MV) =

PROOF. Since (G)»n is always a subalgebra of M({G}MvJ, we have thatB ( ( G ) A O is a subalgebra of B((G)MVJ.

We will see that B((G)MVJ c B((G)MJ- The case G = 0 is clear. Supposethat G is a finite set of cardinality p > 1. Since MVn-algebras are locally finite (see[9, Chapter II, Theorem 10.16]), we obtain that {G)Mvn is a finite MVn-algebra. Sincefinite MVn-algebras are direct products of simple algebras, there exists a finite k > 1such that (G) xv n = n,=i Lm,, where each nij — 1 divides n —1, for each/ = l,...,k.If k = 1, then {G)M,, and (G)MV*

aTe finite chains whose only Boolean elements aretheir extremes. Otherwise, we can think of the elements of (G)MV,, as k-tuples, thatis, if x e (G)MV»> then x = (xu ..., xk). We shall denote by V the k-tuple given by

if i = j ,

It is clear that for each j = 1, . . . , k, V \s in {G)MV.- From this it follows that forevery pair /' j^ j , i, j e {1, . . . , k], there exists an element x e G such that xj ^ x{.Indeed, suppose on the contrary that there exist /, j < k such that x, = Xj, for everyx e G. Then for every z e (G)MV,

w e would have Zj = Zi contradicting the fact that1' is in (G)MVfi.

To see that every Boolean element in (G)MV. ls a l s o m

(G)JU,,, it is enough toprove that lj is in (G)Ma for every j = 1 , . . . , k. For a fixed j , for each / ^ j ,


i = 1 , . . . k, we choose x' e G such that x'j j£ x\. Let ;, be the numerator of x'j e Ln.It is not hard to verify that

k

V = / \ Jh{x').

Therefore V e (G)M, and B((G)MV,) c B({G)Mn).If G is not finite, let y be a Boolean element in (G)MVn. Hence, there exists a finite

subset Gy of G such that y belongs to the subalgebra of (G)Mv, generated by G r

Therefore, since y is Boolean, y belongs to the subalgebra of {G)M. generated by Gy,and we conclude that B((G)MvJ Q B({G)MJ for all sets G. •

Given an algebra A in a variety K, a subalgebra S of A, and an element JC e A, weshall denote by (S, x)/c the subalgebra of A generated by the set 5 U {JC} in K..

LEMMA 2.10. Let C be in Mn and x € C. Let S be a subalgebra ofC such thato?{x) belongs to B(S) for each i = 1 , . . . , n - 1. Then B((S,x)MJ = B(S).

PROOF. Clearly B(S) is a subalgebra of B((S, x)MJ- It is left to check thatB({S, X)M.) 9 B(S). To achieve this aim, we shall study the form of the elements in(S, X)M.- We define for each s e S,

a(s) — s A x,

y, (5) = s A or" (*), for / = 1 , . . . « - 1,

8j(s) = s A -CT,"(JC), for j = 1, . . .n - 1.

For all s e S we have that y, (s) and 8, {s) are in 5 for i = 1 , . . . , n — 1. Let

M : = j y = V A ^(5') : /. e {«,/3, y i , 5 , , ... yB_,,5B_,} and s, e S

We shall see that (S, JC)^ , = M = (M, A, V, - , CT,", . . . , < _ , , 0, 1). Indeed, for all5 € 5, s = y,(s) v5 , ( i ) , and thenS c M. Besides, JC e M because JC = a ( l ) . Lastly,it is easy to see that M is closed under the operations of H-valued Moisil algebra.Thus (S, x)M» is a subalgebra of M. From the definition of M, it is obvious thatM c (S, x)M^ and the equality follows.

Now let z e S((S, JC)M,). By Theorem 2.6, <_,(£) = z and z = V*:=, A,=i /(*.•)

with / € {a, /J, )/|, S | , . . . , yn_i, 5n_i} and s, e 5. Then we have

z = a;_,(Z) = <c, ( V / \ / ( . , ) ) = V/\<-.(/•(*.•)).


is in B(S) because o-;_, (/,(*,-)) = y t ( C f e ) ) or CT;_,(/;(S,)) = «*(CTB"_,(*,-)), for

some k = I,... ,n — 1. D

THEOREM 2.11. Let C be an MVn-algebra and x € C. Le? S be a subalgebra ofCsuch that (T"(X) belongs to B(S)for each i = 1,... ,n — 1. Then

PROOF. ByLemma2.9andLemma2.10weobtainB((S, JC)A1V,,) = B((S, JC)XB) =B(S). n

2.2. Boolean elements in FreeMVii(Z) Recall that a Boolean algebra B is said tobe free over a poset Y if for each Boolean algebra C and for each non-decreasingfunction / : Y -> C, / can be uniquely extended to a homomorphism from B into C.

THEOREM 2.12. B(Free>fv,, (Z)) is the free Boolean algebra over the poset Z' :={or,"(z) : z e Z , i = 1 , . . . , I I - 1}.

PROOF. Let S be the subalgebra of B (Freely,, (Z)) generated by Z'. Let C be aBoolean algebra and let / : Z' —> C be a non-decreasing function. The monotonicityof / implies that the prescription

defines a function f : Z —*• C1"1. Since C1"1 € AiVn, there is a unique homo-morphism h' : Freexv. (Z) -> C'"1 such that h'(z) = f'(z) for every z € Z. Let7r : C1"1 —> C be the projection over the first coordinate. The composition ix oh!restricted to S is a homomorphism h : S —• C, and for y = <r"(z) € Z' we have

Hence S is the free Boolean algebra over the poset Z'. However, since crj(z) is in Sfor all z € Z and y — 1, . . . n — 1, Theorem 2.11 asserts that

for every z e Z. From the fact that S is a subalgebra of B(Free.Mv,r(Z)) we obtain

S = B«S, Z)MVJ = V(FneMv.(Z))

that is, B(Free.v(v,, (Z)) is the free Boolean algebra generated by the poset Z'. •


From Theorem 2.4 we obtain the follwoing result.

COROLLARY 2.13. B(Freev(X)) is the free Boolean algebra generated by theposetY := {or/1 (-.-a) : x € X, i = \,...,n- 1}.

REMARK 2.14. If n — 2, that is, the variety considered V is generated by a BL2-chain, then o}(x) = x for each x e X. Therefore, in this case, Y = {-"-a : x e X],and the cardinality of Y equals the cardinality of X. It follows that B(Freev(X)) isthe free Boolean algebra over the set Y.

3. Fre*v(X)/{U)

Following the program established at the end of Section 2, our next aim is todescribe Freev(X)/(U) for each ultrafilter U in the free Boolean algebra generatedby Y = {a"(-'->x) : x e X, i — 1, . . . , n — 1}, where (U) is the BL-filter generatedby the Boolean filter U.

The plan is to prove that MV(Freev(X)/({/)) is a subalgebra of Ln and then,using Theorem 1.5, decompose each quotient Freev(X)/(U) into an ordinal sum. Toaccomplish this we need the following results.

THEOREM 3.1. Let A be a BL-algebra and U e Sp B(A). Then

fi MV(A)).

PROOF. Let V =: (U) n MV(A) and let / : MV(A)/ V -> MV(A/(f/» be givenby/ (a /V) = a/(t/),foreacha 6 MV(A). It is easy to see that / is a homomorphisminto MV(A/(f/». We have that

(1) fis injective.

Let a/{U) = b/(U), with a, b € MV(A). From Lemma 1.3, we know that thereexists u e U such that a A M = b A u. Since U c M V(A), we have that u e V. Fromthe fact that u is Boolean (see [17, Lemma 2.2]), we have that a *u = a/\u = bAu < b,thus u < a -> b and similarly u < b —>• a. Then a —> b and b —*• a are in V and thismeans that a/V=b/V.

(2) fis surjective.

Leta/((/> e MV(A/((/». Then a/(U) = -<-(a/(U)) = -•-.«/{(/), and since-^-•a € MV(A) we obtain that / ( — a / V ) = a/(f/). D

By Theorem 2.4, if {/ e SpB(Freev(X)), then U is an ultrafilter in


Moreover, {U) n MV(Freev(X)) = (U) n Freely,, (^-X) is the Stone ultrafilter ofFreely, (-•-'A') generated by U. From [14, Chapter 6.3], we have that

(£/>n Freely, ( -"X)

is a maximal filter of Freely,, ( - "X) . It follows from [14, Corollary 3.5.4] that theMV-algebra MV(Freev(X))/((f/) D MV(Freev(X))) is an MV-chain in MVn, thusfrom Theorem 3.1 we have the follwoing result.

THEOREM 3.2. MV(Freev(X)/(£/)) = Ls with s - 1 dividing n - 1.

From Theorems 1.5 and 3.2 we obtain the next result.

THEOREM 3.3. For each U £ SpB(Freev(X)), we have that

Freev(X)/(U) = LsVD(Freev(X)/(U))

for some s — 1 dividing n — 1.

In order to complete the description of Freey(X) we have to find a description ofD(Freev(X)/(£/)) for each U 6 SpB(Freev(A:)). This last description depends onthe characterization of the variety W of generalized BL-algebras generated by thegeneralized BL-chain B. Therefore, we shall firstly consider such variety.

3.1. The subvariety of QBC generated by B We recall that V is the variety of BL-algebras generated by the BL-chain Tn = Ln l±l B. Let W be the variety of generalizedBL-algebras generated by the chain B.

Let {en i e 1} be the set of equations that define MVn as a subvariety of B£,and {dj, j 6 J] be the set of equations that define W as a subvariety of QBC.For each / e / , let e\ be the equation that results from substituting -<->x for eachvariable x in e,, and for each j e J, let d'j be the equation that results from substi-tuting ->->jt —• x for each variable x in the equation dj. Let V denote the varietyof BL-algebras characterized by the equations of BL-algebras plus the equations

{*;, / € / } u {</;., j ej).

THEOREM 3.4. V c V.

PROOF. Let A be a subdirectly irreducible BL-algebra in V. From Theorem 1.1, Ais a BL-chain, and by Theorem 1.4, A = MV(A)WD(A). Since for each x <= MV(A),we have ->-<x = x, MV(A) satisfies equations [et, i e I). Then MV(A) is a chainin MVn, that is, MV(A) = Lv, with s — 1 dividing n — 1. Moreover, since for eachx e D(A), we have -•->.* —• x = x, D(A) satisfies equations {dj, j € J}. Hence


D(A) = C is a generalized BL-chain in W. Since A is subdirectly irreducible, Cis also subdirectly irreducible, and since QBC is a congruence distributive variety,we can apply Jonsson's Lemma (see [9]) to conclude that C e HSPU(B). Hencethere is a set J ^ 0 and an ultrafilter U over J such that C is a homomorphic imageof a subalgebra of BJ/U. From the proof of [2, Proposition 3.3] it follows that(L,, W B)J/U = LJJU l±l BJ/U, and since Ln is finite, VjU = Ln. Now it is easyto see that A = L, W C e HSPu(Ln W B) c V. •

The next corollary states the main result of this section.

COROLLARY 3.5. The variety W of generalized BL-algebras generated by B con-sists of the generalized BL-algebras C such that L , W C belongs to V.

PROOF. Given C e W, Ln l±) C € V c V. On the other hand, if C is a generalizedBL-algebra such that Ln ttl C e V, then the elements of C satisfy equations d'j for eachj 6 J and since ->->x —> x = x for each x € C, the elements of C satisfy equationsdj for each j G l Hence C is in W. •

3.2. D(F ree v (^ ) / ( t / ) ) We know that the ultrafilters of a Boolean algebra arein bijective correspondence with the homomorphisms from the algebra into the twoelements Boolean algebra, 2. Since every upwards closed subset of the poset Y ={o"(->-'x) : x 6 X,i = 1, . . . , n — 1} is in correspondence with an increasingfunction from Y onto 2, and every increasing function from Y can be extended toa homomorphism from B(Freey(X)) into 2, the ultrafilters of B(Freev(X)) are incorrespondence with the upwards closed subsets of Y. This is summarized in thefollowing lemma.

LEMMA 3.6. Consider the poset Y = [ol'(->->x) : x € X, i = 1, . . . , n — 1}.The correspondence that assigns to each upwards closed subset S C Y the Booleanfilter Us generated by the set S U {->y : y € Y \ S}, defines a bijection from the set ofupwards closed subsets ofY onto the ultrafilters o/B(Freey(X)).

We shall refer to each member of Sp B(Freev(X)) by Us making explicit referenceto the upwards closed subset S that corresponds to it.

LEMMA 3.7. Let F s be the subalgebra of the generalized BL-algebra

D(Freev(X)/({/5»

generated by the set Xs :— \x/{Us) : x € X, -.-uc 6 (Us)}- Then

F5 = D(Freev(X)/({/s>).


PROOF. Freev(X)/{£/s) is the BL-algebra generated by the set Zs = [x/{Us) :x € X}. From Theorem 3.3, there exists an integer m such that

Freev(X)/(t/s) = Lm W D(Freev(X)/(£/s)).

Hence each element of Zs is either in Lm \ {T} or it is in D(Freey(X)/(i/s))-If Xs = 0, then Fs = D(Freev(X)/(£/s» = {T}. So let us suppose Xs £ 0. Let

y € D(Freev(X)/(f/s)). Recalling that F s is the generalized BL-algebra generatedby Xs, we will check that y is in Fs. Since y e Freey(X)/(£/s), y is given by a termon the elements x/(Us) e Zs. By induction on the complexity of y, we have:

• If y is a generator, that is, y = x/(Us) for some x/(Us) € Zs, since y €D(Freev(X)/(f/s}), we have that T = — y = ->^(x/(Us)) = (-^x)/(Us).This happens only if ->->JC e (Us).

• Suppose that for each element z e D(Freev(X)/{Us)) of complexity lessthan k, z can be written as a term in the variables x/(Us) in Xs. Let y eZ)(Freev(Ar)/(f/s)) be an element of complexity Ic. The possible cases arethe following:

(1) y = a —> b for some elements a, b of complexity < k. In this case thepossibilities are

(a) a <b. This means a -> ft = T and y can be written as x/(Us) ->•x/(Us) for any x/(Us) € Xs, and thus y 6 Fs,

(b) a £ b. Since y = a ->• ft is in D(Freev(X)/(f/s)), the only possi-bility is that a,b € D(Freev(X)/(Us)) and by inductive hypothesisy is in Fs.

(2) y = a * b for some elements a, ft of complexity < k. In this casenecessarily a, ft e D(Freev(X)/(C/s)) and by inductive hypothesis y isin F5.

Then for each y e D(Freev(X)/(f/s>), y can be written as a term on the elementsof Xs. Therefore y 6 Fs and we conclude that Fs = D(Freev(X)/(Us)). •

With the notation of the previous lemma, we have the following theorem.

THEOREM 3.8. For each Us in SpB(Freev(X))(

D(Freev(X)/(t/s)) £ Freew(Xs).

PROOF. From Theorem 2.6 and Lemma 3.6 we can deduced that ->->x e (Us) ifand only ifa?(-^-^x) e S if and only if a," (->->x) e S for i = 1 , . . . , n - 1. Hence if-'-•jr ^ ((/s) there is a j such that o"(->-<x) <£ S. We define, for each x e X,

if — J t e (t/s),

-•x) & S] otherwise.max{j € {1, n - 1 ) :


Let C 6 W and let C = Ln l±l C. From Theorem 3.5, C is in V. Given a function

/ : Xs —• C, define / : X —> C by the prescriptions:

if — * e

[in — jx — l ) / (« — 1) otherwise.

There is a unique homomorphism h : Freey(X) —> C such that hix) =for each x e X. We have that Us £ / r ' ({T}) . Indeed, if ->->JC 6 (£/s), thenhia"{—'—'x)) = cj'

1(—>—'(/J(JC)) = <T"(—>—'fixf(Us))) = CT"(T) = T. If —•—>jc ̂ {Us),then

I -L if ' < jx,

T otherwise.

Hence there is a unique homomorphism hx : Freev(X)/(f/5) -> C such thathi(a/(Us)) = Ha) for all a e Freev(X). By Lemma 3.7, D(Freev(X)/{f/s)) isthe algebra generated by Xs. Then the restriction h of /z, to D(Freev(A')/({/s)) is ahomomorphism h : D(Freey(X)/({/$)) ->• C, and for each x such that ->->* e (t/s).

h(x/(Us)) = hdx/{Us)) = Hx) = fix) = fix / {Us)).

Therefore we conclude that D(Freev(Ar)/({/s» = FreewtX*;). •

THEOREM 3.9. The free BL-algebra FreeyfXj can be represented as a weakBoolean product of the family (Freev(X)/(f/5>) : Us € SpB(Freev(X)), whereB(Freey(A')) is the free Boolean algebra over the poset Y = {a"(->->x) : x e X,i = 1, . . . , « — 1}. Moreover, for each Us € SpB(Freey(X)), there exists m > 2such that m — 1 divides n — \ and

¥nev(X)/(Us) = Lm W FreewiXs),

whereXs '•= {x/(Us) '• —•~ix e (Us)} andW is the variety of generalized BL-algebrasgenerated by B.

4. Examples

4.1. PL-algebras Let G be a lattice-ordered abelian group (£-group), and G~ ={x e G : x < 0} its negative cone. For each pair of elements x, y € G~, we definethe following operators:

x * y = x + y and x-^ y = 0 A iy - x).

Then G~ = (G~, A, V, *, ->, 0) is a generalized BL-algebra. The following result

can be deduced from [3] (see also [6] and [15]).


THEOREM 4.1. The following conditions are equivalent for a generalized BL-algebra A:

(1) A is a cancellative hoop.(2) There is an l-group G such that A ~ G".(3) A is in the variety of generalized BL-algebras generated by Z~, where Z denotes

the additive group of integers with the usual order.

Let us consider W, the variety of generalized BL-algebras generated by Z", thatis, the variety of cancellative hoops. In [16] a description of Free w (X) is given forany set X of free generators. Therefore we can have a complete description of freealgebras in varieties of BL-algebras generated by the ordinal sum

P L h = L B W Z - .

Indeed, if we denote by VCn the variety of BL-algebras generated by PL,,, fromTheorem 3.9 we obtain that F r e e ^ C X ) is a weak Boolean product of algebras ofthe form Ls l±l Freew(X') with s — 1 dividing n — 1 and some set X' of cardinalityless or equal than X. Therefore, in the present case, the BL-algebra Free-pcA^) c a n

be completely described as a weak Boolean product of ordinal sums of two knownalgebras.

From [15, Theorem 2.8], VC2 is the variety of PL-algebras VC. From Remark 2.14,SpBCFreepcCX)) is the Cantor space 2 m . From Theorem 3.9, the free PL-algebraover a set X can be describe as a weak Boolean product over the Cantor space 2|X|

of algebras of the form L2 t±l Freew(X') for some set X' of cardinality less or equalthan X.

Given a BL-algebra A, the radical R(\) of A is the intersection of all maximalimplicative filters of A. We have that r(A) = (/?(A), *, —>, A , V, T) is a generalizedBL-algebra. Let

VC = (R : R = r(A) for some A g VC).

VC is a variety of generalized BL-algebras. In [17] a description of Free-p,c(X) isgiven. From Example 4.7 and Theorem 5.7 in the mentioned paper we obtain thatFreeP £(X) is the weak Boolean product of the family (L2 l±l FreeVC'(S) : S c 2|Af|)over the Cantor space 2|X|. In order to check that our description and the one given in[17] coincide it is only left to check that VC = W. From Corollary 3.5 we have thatW consist on the generalized BL-algebras C such that L2 W C e VC.

THEOREM 4.2. VC = W.

PROOF. LetC g VC. Then there exists a BL-algebra A e VC such that r(A) = C.It is not hard to check that L2 l±l C is a subalgebra of A, thus L2 W C is in VC. It


follows that C 6 W. On the other hand, let C e W. Then L2 W C is in VC, andC e VC. •

4.2. Finitely generated free algebras As we mentioned in the introduction, whenthe set of generators X is finite, let us say of cardinality k, the algebra Freev(X) isdescribed in [10] asadirect product of algebras of the form L(WFreew(X'), withs — 1that divides n - 1 and some set X' of cardinality less than or equal to the cardinalityof X, where W is again the subvariety of QBC generated by B. The method used todescribe the algebras strongly relies on the fact that the Boolean elements of Freey(X)form a finite Boolean algebra. Indeed, Freev(X) is a direct product of nk algebrasobtained by taking the quotients by the implicative filters generated by the atoms ofB(Freey(X)). In this case, once you know the form of the atom that generates theultrafilter U you also know the number s such that MV((Freev(X))/(£/)) = Ls..

When the set X of generators is finite, of cardinality k, then Y = {a"(-'->x) :x € X, i — 1, . . . , n — 1} is the cardinal sum of k chains of length n — 1. Thereforethe number of upwards closed subsets of Y is nk. Since weak Boolean products overdiscrete finite spaces coincide with direct products, Theorem 3.9 asserts that Freev(X)is a direct product of nk BL-algebras of the form L, W Freew(K), with s — 1 thatdivides n — 1 and some set Y of cardinality less than or equal to k.

Therefore the description given in the present paper coincides with the one in [10].However, the description given in [10], based on a detailed analysis of the structure ofthe atoms of B(Freev(X)) for a finite X, is more precise because it gives the numberof factors of each kind appearing in the direct product representation.

References

[1] P. Agliano, I. M. A. Ferreirim and F. Montagna, 'Basic hoops: An algebraic study of continuoust-norms', manuscript.

[2] P. Agliano and F. Montagna, 'Varieties of BL-algebras I: General properties', J. Pure Appl. Algebra181 (2003), 105-129.

[3] K. Amer, 'Equationally complete classes of conmutative monoids with monus', Algebra Univer-ia/j.918(1984), 129-131.

[4] R. Balbes and P. Dwinger, Distributive lattices (University of Missoury Press, Columbia, 1974).[5] D. Bigelow and S. Burris, 'Boolean algebras of factor congruences', Ada Sci. Math. (Szeged) 54

(1990), 11-20.[6] W. J. Block and I. M. A. Ferreirim, 'Hoops and their implicational reducts (Abstract)', in: Algebraic

Methods in Logic and Computer Sciences, Banach Center Publications 28 (Polish Academy ofScience, Warsaw, 1993) pp. 219-230.

[7] , 'On the structure of hoops', Algebra Universal 43 (2000), 233-257.[8] V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu, Lukasiewicz-Moisil algebras (Elsevier,

Amsterdam, 1991).[9] S. Burris and H. P. Sankappanavar, A course in universal algebra (Springer, New York, 1981).


[10] M. Busaniche, 'Free algebras in varieties of BL-algebras generated by a chain', Algebra Universalis50 (2003), 259-277.

[11] R. Cignoli, Moisil algebras, Notas de logica matemdtica (Instituto De Matematica, UniversidadNac. del Sur, Bahia Blanca, Argentina, 1970).

[12] , 'Some algebraic aspects of many-valued logics', in: Proceedings of the third BrazilianConference on Mathematical Logic (Sociedade Brasileira de Logica) (eds. A. I. Arruda, N. C. A.da Costa and A. M. Sette) (Sao Paulo, 1980) pp. 49-69.

[13] , 'Proper n-valued tukasiewicz algebras as S-algebras of Jukasiewicz n-valued propositionalcalculi', Studia Logica 41 (1982), 3-16.

[14] R. Cignoli, M. I. D'Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning(Kluwer, Dordrecht, 2000).

[15] R. Cignoli and A. Torrens, 'An algebraic analysis of product logic', Mult.-Valued Log. 5 (2000),45-65.

[16] , 'Free cancelative hoops', Algebra Universalis 43 (2000), 213-216.[17] , 'Free algebras in varieties of BL-algebras with a Boolean retract', Algebra Universalis 48

(2002), 55-79.[18] , 'Hajek basic fuzzy logic and lukasiewicz infinite-valued logic', Arch. Math. Logic 42

(2003), 361-370.[19] P. Hajek, Metamathematics of fuzzy logic (Kluwer, Dordrecht, 1998).[20] A. Horn, 'Free L-algebras', J. Symbolic Logic 34 (1969), 475-480.[21] A. Iorgulescu, 'Connections between MVn-algebras and n-valued lukasiewicz-moisil algebras.

Part I', Discrete Math. 181 (1998), 155-177.[22] R. McNaughton, 'A theorem about infinite-valued sentential logic', J. Symbolic Logic 16 (1951),

1-13.[23] A. Di Nola, G. Georgescu and L. Leustean, 'Boolean products of BL-algebras', J. Math. Anal.

Appl. 251(2000), 106-131.[24] A. J. Rodriguez and A. Torrens, 'Wajsberg algebras and post algebras', Studia Logica 53 (1994),

1-19.[25] J. von Plato, 'Skolem's discovery of Godel-Dummett logic', Studia Logica 73 (2003), 153-157.

Departamento de MatematicaFacultad de Ciencias Exactas y NaturalesUniversidad de Buenos Aires - CONICETCiudad Universitaria, 1428 Buenos AiresArgentinae-mail: [email protected], [email protected]

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